Construct Single-Hierarchical P/NBD Model for Long Timeframe Synthetic Data
In this workbook we construct the non-hierarchical P/NBD models on the synthetic data with the longer timeframe.
1 Load and Construct Datasets
We start by modelling the P/NBD model using our synthetic datasets before we try to model real-life data.
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use_fit_start_date <- as.Date("2010-01-01")
use_fit_end_date <- as.Date("2022-01-01")
use_valid_start_date <- as.Date("2022-01-01")
use_valid_end_date <- as.Date("2023-01-01")1.1 Load Long Time-frame Synthetic Data
We now want to load the short time-frame synthetic data.
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customer_cohortdata_tbl <- read_rds("data/synthdata_longframe_cohort_tbl.rds")
customer_cohortdata_tbl |> glimpse()Rows: 50,000
Columns: 4
$ customer_id <chr> "LFC201001_0001", "LFC201001_0002", "LFC201001_0003", "…
$ cohort_qtr <chr> "2010 Q1", "2010 Q1", "2010 Q1", "2010 Q1", "2010 Q1", …
$ cohort_ym <chr> "2010 01", "2010 01", "2010 01", "2010 01", "2010 01", …
$ first_tnx_date <date> 2010-01-01, 2010-01-01, 2010-01-01, 2010-01-01, 2010-0…
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customer_simparams_tbl <- read_rds("data/synthdata_longframe_simparams_tbl.rds")
customer_simparams_tbl |> glimpse()Rows: 50,000
Columns: 9
$ customer_id <chr> "LFC201001_0001", "LFC201001_0002", "LFC201001_0003", …
$ cohort_qtr <chr> "2010 Q1", "2010 Q1", "2010 Q1", "2010 Q1", "2010 Q1",…
$ cohort_ym <chr> "2010 01", "2010 01", "2010 01", "2010 01", "2010 01",…
$ first_tnx_date <date> 2010-01-01, 2010-01-01, 2010-01-01, 2010-01-01, 2010-…
$ customer_lambda <dbl> 6.349657e-02, 1.699536e-01, 4.675286e-02, 4.760263e-02…
$ customer_mu <dbl> 0.243178098, 0.122825722, 0.049332886, 0.007878287, 0.…
$ customer_tau <dbl> 6.6538597, 9.3140562, 99.3492910, 171.9080177, 0.76405…
$ customer_amtmn <dbl> 168.410136, 90.347217, 32.472693, 117.367925, 70.40242…
$ customer_amtcv <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
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customer_transactions_tbl <- read_rds("data/synthdata_longframe_transactions_tbl.rds")
customer_transactions_tbl |> glimpse()Rows: 461,430
Columns: 4
$ customer_id <chr> "LFC201001_0003", "LFC201001_0002", "LFC201001_0004", "L…
$ tnx_timestamp <dttm> 2010-01-01 08:49:10, 2010-01-01 10:00:52, 2010-01-01 11…
$ invoice_id <chr> "T20100101-0001", "T20100101-0002", "T20100101-0003", "T…
$ tnx_amount <dbl> 5.80, 84.34, 6.71, 17.90, 98.92, 91.74, 122.70, 198.48, …
1.2 Load Derived Data
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id_1000 <- read_rds("data/longsynth_id_1000.rds")
id_5000 <- read_rds("data/longsynth_id_5000.rds")
id_10000 <- read_rds("data/longsynth_id_10000.rds")
fit_1000_data_tbl <- read_rds("data/longsynth_fit_1000_data_tbl.rds")
fit_10000_data_tbl <- read_rds("data/longsynth_fit_10000_data_tbl.rds")
customer_fit_stats_tbl <- fit_1000_data_tbl
customer_summarystats_tbl <- read_rds("data/longsynth_customer_summarystats_tbl.rds")
obs_fitdata_tbl <- read_rds("data/longsynth_obs_fitdata_tbl.rds")
obs_validdata_tbl <- read_rds("data/longsynth_obs_validdata_tbl.rds")Finally, we need to set our directories where we save our Stan code and the model outputs.
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stan_modeldir <- "stan_models"
stan_codedir <- "stan_code"2 Fit First Hierarchical Lambda Model
Our first hierarchical model puts a hierarchical prior around the mean of our population \(\lambda\) - lambda_mn.
Once again we use a Gamma prior for it.
2.1 Compile and Fit Stan Model
We now compile this model using CmdStanR.
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pnbd_onehierlambda_stanmodel <- cmdstan_model(
"stan_code/pnbd_onehier_lambda.stan",
include_paths = stan_codedir,
pedantic = TRUE,
dir = stan_modeldir
)We then use this compiled model with our data to produce a fit of the data.
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stan_modelname <- "pnbd_long_onehierlambda1"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed <- stanfit_seed + 1
stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")
stan_data_lst <- customer_fit_stats_tbl |>
select(customer_id, x, t_x, T_cal) |>
compose_data(
hier_lambda_mn_p1 = 0.25,
hier_lambda_mn_p2 = 1,
lambda_cv = 1.00,
mu_mn = 0.10,
mu_cv = 1.00,
)
if(!file_exists(stanfit_object_file)) {
pnbd_long_onehierlambda1_stanfit <- pnbd_onehierlambda_stanmodel$sample(
data = stan_data_lst,
chains = 4,
iter_warmup = 500,
iter_sampling = 500,
seed = stanfit_seed,
save_warmup = TRUE,
output_dir = stan_modeldir,
output_basename = stanfit_prefix,
)
pnbd_long_onehierlambda1_stanfit$save_object(stanfit_object_file, compress = "gzip")
} else {
pnbd_long_onehierlambda1_stanfit <- read_rds(stanfit_object_file)
}
pnbd_long_onehierlambda1_stanfit$summary()# A tibble: 3,003 × 10
variable mean median sd mad q5 q95 rhat ess_bulk
<chr> <num> <num> <num> <num> <num> <num> <num> <num>
1 lp__ -2.05e+4 -2.05e+4 33.9 35.8 -2.06e+4 -2.05e+4 1.01 688.
2 lambda_mn 2.43e-1 2.42e-1 0.0102 0.0101 2.26e-1 2.59e-1 1.00 2886.
3 lambda[1] 2.08e-1 1.64e-1 0.172 0.133 2.91e-2 5.50e-1 1.00 3491.
4 lambda[2] 1.37e-1 8.13e-2 0.164 0.0934 4.78e-3 4.47e-1 1.00 2523.
5 lambda[3] 5.94e-2 4.77e-2 0.0464 0.0379 9.02e-3 1.46e-1 0.999 3943.
6 lambda[4] 1.38e-1 8.34e-2 0.161 0.0925 5.52e-3 4.59e-1 1.00 2692.
7 lambda[5] 4.83e-1 4.63e-1 0.164 0.154 2.56e-1 7.82e-1 1.00 5276.
8 lambda[6] 3.00e-1 2.35e-1 0.241 0.192 4.38e-2 7.82e-1 1.00 5051.
9 lambda[7] 1.35e-1 8.00e-2 0.162 0.0884 6.24e-3 4.46e-1 1.00 3138.
10 lambda[8] 1.44e-1 8.43e-2 0.173 0.0962 5.90e-3 4.78e-1 1.00 3325.
# ℹ 2,993 more rows
# ℹ 1 more variable: ess_tail <num>
We have some basic HMC-based validity statistics we can check.
Show code
pnbd_long_onehierlambda1_stanfit$cmdstan_diagnose()Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehierlambda1-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehierlambda1-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehierlambda1-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehierlambda1-4.csvWarning: non-fatal error reading adaptation data
Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.
Checking sampler transitions for divergences.
No divergent transitions found.
Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.
Effective sample size satisfactory.
Split R-hat values satisfactory all parameters.
Processing complete, no problems detected.
2.2 Visual Diagnostics of the Sample Validity
Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.
Show code
parameter_subset <- c(
"lambda_mn",
"lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
"mu[1]", "mu[2]", "mu[3]", "mu[4]"
)
pnbd_long_onehierlambda1_stanfit$draws(inc_warmup = FALSE) |>
mcmc_trace(pars = parameter_subset) +
expand_limits(y = 0) +
labs(
x = "Iteration",
y = "Value",
title = "Traceplot of Sample of Lambda and Mu Values"
) +
theme(axis.text.x = element_text(size = 10))We also check \(N_{eff}\) as a quick diagnostic of the fit.
Show code
pnbd_long_onehierlambda1_stanfit |>
neff_ratio(pars = c("lambda", "mu")) |>
as.numeric() |>
mcmc_neff() +
ggtitle("Plot of Parameter Effective Sample Sizes")2.3 Assess the Model
As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.
Show code
pnbd_long_onehierlambda1_assess_data_lst <- run_model_assessment(
model_stanfit = pnbd_long_onehierlambda1_stanfit,
insample_tbl = customer_fit_stats_tbl,
outsample_tbl = customer_valid_stats_tbl,
fit_label = "pnbd_long_onehierlambda1",
fit_end_dttm = use_fit_end_date |> as.POSIXct(),
valid_start_dttm = use_valid_start_date |> as.POSIXct(),
valid_end_dttm = use_valid_end_date |> as.POSIXct(),
sim_seed = 4210
)
pnbd_long_onehierlambda1_assess_data_lst |> glimpse()List of 3
$ model_simstats_filepath : 'glue' chr "data/pnbd_long_onehierlambda1_assess_model_simstats_tbl.rds"
$ model_fit_simstats_filepath : 'glue' chr "data/pnbd_long_onehierlambda1_assess_fit_simstats_tbl.rds"
$ model_valid_simstats_filepath: 'glue' chr "data/pnbd_long_onehierlambda1_assess_valid_simstats_tbl.rds"
2.3.1 Check In-Sample Data Validation
We first check the model against the in-sample data.
Show code
simdata_tbl <- pnbd_long_onehierlambda1_assess_data_lst |>
use_series(model_fit_simstats_filepath) |>
read_rds()
insample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_fitdata_tbl,
simdata_tbl = simdata_tbl
)
insample_plots_lst$multi_plot |> print()Show code
insample_plots_lst$total_plot |> print()Show code
insample_plots_lst$quant_plot |> print()This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.
2.3.2 Check Out-of-Sample Data Validation
We now repeat for the out-of-sample data.
Show code
simdata_tbl <- pnbd_long_onehierlambda1_assess_data_lst |>
use_series(model_valid_simstats_filepath) |>
read_rds()
outsample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_validdata_tbl,
simdata_tbl = simdata_tbl
)
outsample_plots_lst$multi_plot |> print()Show code
outsample_plots_lst$total_plot |> print()Show code
outsample_plots_lst$quant_plot |> print()As for our short time frame data, overall our model is working well.
3 Fit Second Hierarchical Lambda Model
In this model, we are going with a broadly similar model but we are instead using a different mean for our hierarchical prior.
3.1 Fit Stan Model
We now want to fit the model to our data using this alternative prior for lambda_mn.
Show code
stan_modelname <- "pnbd_long_onehierlambda2"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed <- stanfit_seed + 1
stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")
stan_data_lst <- customer_fit_stats_tbl |>
select(customer_id, x, t_x, T_cal) |>
compose_data(
hier_lambda_mn_p1 = 0.50,
hier_lambda_mn_p2 = 1,
lambda_cv = 1.00,
mu_mn = 0.10,
mu_cv = 1.00,
)
if(!file_exists(stanfit_object_file)) {
pnbd_long_onehierlambda2_stanfit <- pnbd_onehierlambda_stanmodel$sample(
data = stan_data_lst,
chains = 4,
iter_warmup = 500,
iter_sampling = 500,
seed = stanfit_seed,
save_warmup = TRUE,
output_dir = stan_modeldir,
output_basename = stanfit_prefix,
)
pnbd_long_onehierlambda2_stanfit$save_object(stanfit_object_file, compress = "gzip")
} else {
pnbd_long_onehierlambda2_stanfit <- read_rds(stanfit_object_file)
}
pnbd_long_onehierlambda2_stanfit$summary()# A tibble: 3,003 × 10
variable mean median sd mad q5 q95 rhat ess_bulk
<chr> <num> <num> <num> <num> <num> <num> <num> <num>
1 lp__ -2.05e+4 -2.05e+4 3.39e+1 34.0 -2.06e+4 -2.05e+4 1.00 636.
2 lambda_mn 2.42e-1 2.42e-1 9.83e-3 0.0105 2.27e-1 2.59e-1 1.00 2485.
3 lambda[1] 2.07e-1 1.64e-1 1.68e-1 0.128 2.74e-2 5.42e-1 1.00 3343.
4 lambda[2] 1.42e-1 8.19e-2 1.71e-1 0.0954 4.46e-3 4.64e-1 1.00 2926.
5 lambda[3] 5.89e-2 4.47e-2 4.79e-2 0.0363 8.35e-3 1.56e-1 1.00 3338.
6 lambda[4] 1.35e-1 8.33e-2 1.50e-1 0.0928 5.63e-3 4.47e-1 1.00 2677.
7 lambda[5] 4.83e-1 4.62e-1 1.66e-1 0.159 2.54e-1 7.84e-1 1.00 4314.
8 lambda[6] 2.99e-1 2.22e-1 2.56e-1 0.192 3.52e-2 7.89e-1 1.00 2924.
9 lambda[7] 1.36e-1 7.97e-2 1.63e-1 0.0922 4.56e-3 4.65e-1 1.00 2782.
10 lambda[8] 1.35e-1 8.05e-2 1.60e-1 0.0901 5.56e-3 4.54e-1 1.00 2829.
# ℹ 2,993 more rows
# ℹ 1 more variable: ess_tail <num>
We have some basic HMC-based validity statistics we can check.
Show code
pnbd_long_onehierlambda2_stanfit$cmdstan_diagnose()Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehierlambda2-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehierlambda2-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehierlambda2-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehierlambda2-4.csvWarning: non-fatal error reading adaptation data
Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.
Checking sampler transitions for divergences.
No divergent transitions found.
Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.
Effective sample size satisfactory.
Split R-hat values satisfactory all parameters.
Processing complete, no problems detected.
3.2 Visual Diagnostics of the Sample Validity
Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.
Show code
parameter_subset <- c(
"lambda_mn",
"lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
"mu[1]", "mu[2]", "mu[3]", "mu[4]"
)
pnbd_long_onehierlambda2_stanfit$draws(inc_warmup = FALSE) |>
mcmc_trace(pars = parameter_subset) +
expand_limits(y = 0) +
labs(
x = "Iteration",
y = "Value",
title = "Traceplot of Sample of Lambda and Mu Values"
) +
theme(axis.text.x = element_text(size = 10))We also check \(N_{eff}\) as a quick diagnostic of the fit.
Show code
pnbd_long_onehierlambda2_stanfit |>
neff_ratio(pars = c("lambda", "mu")) |>
as.numeric() |>
mcmc_neff() +
ggtitle("Plot of Parameter Effective Sample Sizes")3.3 Assess the Model
As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.
Show code
pnbd_long_onehierlambda2_assess_data_lst <- run_model_assessment(
model_stanfit = pnbd_long_onehierlambda2_stanfit,
insample_tbl = customer_fit_stats_tbl,
outsample_tbl = customer_valid_stats_tbl,
fit_label = "pnbd_long_onehierlambda2",
fit_end_dttm = use_fit_end_date |> as.POSIXct(),
valid_start_dttm = use_valid_start_date |> as.POSIXct(),
valid_end_dttm = use_valid_end_date |> as.POSIXct(),
sim_seed = 4210
)
pnbd_long_onehierlambda2_assess_data_lst |> glimpse()List of 3
$ model_simstats_filepath : 'glue' chr "data/pnbd_long_onehierlambda2_assess_model_simstats_tbl.rds"
$ model_fit_simstats_filepath : 'glue' chr "data/pnbd_long_onehierlambda2_assess_fit_simstats_tbl.rds"
$ model_valid_simstats_filepath: 'glue' chr "data/pnbd_long_onehierlambda2_assess_valid_simstats_tbl.rds"
3.3.1 Check In-Sample Data Validation
We first check the model against the in-sample data.
Show code
simdata_tbl <- pnbd_long_onehierlambda2_assess_data_lst |>
use_series(model_fit_simstats_filepath) |>
read_rds()
insample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_fitdata_tbl,
simdata_tbl = simdata_tbl
)
insample_plots_lst$multi_plot |> print()Show code
insample_plots_lst$total_plot |> print()Show code
insample_plots_lst$quant_plot |> print()This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.
3.3.2 Check Out-of-Sample Data Validation
We now repeat for the out-of-sample data.
Show code
simdata_tbl <- pnbd_long_onehierlambda1_assess_data_lst |>
use_series(model_valid_simstats_filepath) |>
read_rds()
outsample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_validdata_tbl,
simdata_tbl = simdata_tbl
)
outsample_plots_lst$multi_plot |> print()Show code
outsample_plots_lst$total_plot |> print()Show code
outsample_plots_lst$quant_plot |> print()As for our short time frame data, overall our model is working well.
4 Fit First Hierarchical Mu Model
We now construct the same hierarchical model but based around mu_mn.
4.1 Compile and Fit Stan Model
We compile this model using CmdStanR.
Show code
pnbd_onehiermu_stanmodel <- cmdstan_model(
"stan_code/pnbd_onehier_mu.stan",
include_paths = stan_codedir,
pedantic = TRUE,
dir = stan_modeldir
)We then use this compiled model with our data to produce a fit of the data.
Show code
stan_modelname <- "pnbd_long_onehiermu1"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed <- stanfit_seed + 1
stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")
stan_data_lst <- customer_fit_stats_tbl |>
select(customer_id, x, t_x, T_cal) |>
compose_data(
hier_mu_mn_p1 = 0.50,
hier_mu_mn_p2 = 1.00,
lambda_mn = 0.25,
lambda_cv = 1.00,
mu_cv = 1.00
)
if(!file_exists(stanfit_object_file)) {
pnbd_long_onehiermu1_stanfit <- pnbd_onehiermu_stanmodel$sample(
data = stan_data_lst,
chains = 4,
iter_warmup = 500,
iter_sampling = 500,
seed = stanfit_seed,
save_warmup = TRUE,
output_dir = stan_modeldir,
output_basename = stanfit_prefix,
)
pnbd_long_onehiermu1_stanfit$save_object(stanfit_object_file, compress = "gzip")
} else {
pnbd_long_onehiermu1_stanfit <- read_rds(stanfit_object_file)
}
pnbd_long_onehiermu1_stanfit$summary()# A tibble: 3,003 × 10
variable mean median sd mad q5 q95 rhat ess_bulk
<chr> <num> <num> <num> <num> <num> <num> <num> <num>
1 lp__ -1.94e+4 -1.94e+4 3.31e+1 3.45e+1 -1.95e+4 -1.94e+4 1.00 732.
2 mu_mn 1.04e-1 1.03e-1 6.57e-3 6.62e-3 9.36e-2 1.15e-1 1.00 1166.
3 lambda[1] 2.12e-1 1.64e-1 1.78e-1 1.41e-1 2.66e-2 5.44e-1 1.00 3972.
4 lambda[2] 1.46e-1 8.58e-2 1.82e-1 9.80e-2 4.68e-3 4.63e-1 1.00 2585.
5 lambda[3] 5.82e-2 4.62e-2 4.48e-2 3.68e-2 9.14e-3 1.46e-1 0.999 3470.
6 lambda[4] 1.41e-1 7.91e-2 1.72e-1 9.02e-2 5.15e-3 4.77e-1 1.00 2849.
7 lambda[5] 4.85e-1 4.69e-1 1.62e-1 1.61e-1 2.55e-1 7.80e-1 1.00 3873.
8 lambda[6] 2.92e-1 2.34e-1 2.29e-1 1.90e-1 4.34e-2 7.48e-1 1.00 3000.
9 lambda[7] 1.43e-1 8.07e-2 1.77e-1 9.44e-2 4.19e-3 4.98e-1 1.00 2690.
10 lambda[8] 1.39e-1 8.60e-2 1.57e-1 9.29e-2 6.13e-3 4.41e-1 1.00 2766.
# ℹ 2,993 more rows
# ℹ 1 more variable: ess_tail <num>
We have some basic HMC-based validity statistics we can check.
Show code
pnbd_long_onehiermu1_stanfit$cmdstan_diagnose()Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehiermu1-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehiermu1-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehiermu1-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehiermu1-4.csvWarning: non-fatal error reading adaptation data
Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.
Checking sampler transitions for divergences.
No divergent transitions found.
Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.
Effective sample size satisfactory.
Split R-hat values satisfactory all parameters.
Processing complete, no problems detected.
4.2 Visual Diagnostics of the Sample Validity
Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.
Show code
parameter_subset <- c(
"mu_mn",
"lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
"mu[1]", "mu[2]", "mu[3]", "mu[4]"
)
pnbd_long_onehiermu1_stanfit$draws(inc_warmup = FALSE) |>
mcmc_trace(pars = parameter_subset) +
expand_limits(y = 0) +
labs(
x = "Iteration",
y = "Value",
title = "Traceplot of Sample of Lambda and Mu Values"
) +
theme(axis.text.x = element_text(size = 10))We also check \(N_{eff}\) as a quick diagnostic of the fit.
Show code
pnbd_long_onehiermu1_stanfit |>
neff_ratio(pars = c("lambda", "mu")) |>
as.numeric() |>
mcmc_neff() +
ggtitle("Plot of Parameter Effective Sample Sizes")4.3 Assess the Model
As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.
Show code
pnbd_long_onehiermu1_assess_data_lst <- run_model_assessment(
model_stanfit = pnbd_long_onehiermu1_stanfit,
insample_tbl = customer_fit_stats_tbl,
outsample_tbl = customer_valid_stats_tbl,
fit_label = "pnbd_long_onehiermu1",
fit_end_dttm = use_fit_end_date |> as.POSIXct(),
valid_start_dttm = use_valid_start_date |> as.POSIXct(),
valid_end_dttm = use_valid_end_date |> as.POSIXct(),
sim_seed = 4210
)
pnbd_long_onehiermu1_assess_data_lst |> glimpse()List of 3
$ model_simstats_filepath : 'glue' chr "data/pnbd_long_onehiermu1_assess_model_simstats_tbl.rds"
$ model_fit_simstats_filepath : 'glue' chr "data/pnbd_long_onehiermu1_assess_fit_simstats_tbl.rds"
$ model_valid_simstats_filepath: 'glue' chr "data/pnbd_long_onehiermu1_assess_valid_simstats_tbl.rds"
4.3.1 Check In-Sample Data Validation
We first check the model against the in-sample data.
Show code
simdata_tbl <- pnbd_long_onehiermu1_assess_data_lst |>
use_series(model_fit_simstats_filepath) |>
read_rds()
insample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_fitdata_tbl,
simdata_tbl = simdata_tbl
)
insample_plots_lst$multi_plot |> print()Show code
insample_plots_lst$total_plot |> print()Show code
insample_plots_lst$quant_plot |> print()This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.
4.3.2 Check Out-of-Sample Data Validation
We now repeat for the out-of-sample data.
Show code
simdata_tbl <- pnbd_long_onehierlambda1_assess_data_lst |>
use_series(model_valid_simstats_filepath) |>
read_rds()
outsample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_validdata_tbl,
simdata_tbl = simdata_tbl
)
outsample_plots_lst$multi_plot |> print()Show code
outsample_plots_lst$total_plot |> print()Show code
outsample_plots_lst$quant_plot |> print()As for our short time frame data, overall our model is working well.
5 Fit Second Hierarchical Lambda Model
In this model, we are going with a broadly similar model but we are instead using a different mean for our hierarchical prior.
5.1 Fit Stan Model
We now want to fit the model to our data using this alternative prior for lambda_mn.
Show code
stan_modelname <- "pnbd_long_onehiermu2"
stanfit_prefix <- str_c("fit_", stan_modelname)
stanfit_seed <- stanfit_seed + 1
stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")
stan_data_lst <- customer_fit_stats_tbl |>
select(customer_id, x, t_x, T_cal) |>
compose_data(
hier_mu_mn_p1 = 0.25,
hier_mu_mn_p2 = 1.00,
lambda_mn = 0.25,
lambda_cv = 1.00,
mu_cv = 1.00
)
if(!file_exists(stanfit_object_file)) {
pnbd_long_onehiermu2_stanfit <- pnbd_onehiermu_stanmodel$sample(
data = stan_data_lst,
chains = 4,
iter_warmup = 500,
iter_sampling = 500,
seed = stanfit_seed,
save_warmup = TRUE,
output_dir = stan_modeldir,
output_basename = stanfit_prefix,
)
pnbd_long_onehiermu2_stanfit$save_object(stanfit_object_file, compress = "gzip")
} else {
pnbd_long_onehiermu2_stanfit <- read_rds(stanfit_object_file)
}
pnbd_long_onehiermu2_stanfit$summary()# A tibble: 3,003 × 10
variable mean median sd mad q5 q95 rhat ess_bulk
<chr> <num> <num> <num> <num> <num> <num> <num> <num>
1 lp__ -1.94e+4 -1.94e+4 3.46e+1 3.56e+1 -1.95e+4 -1.94e+4 1.00 677.
2 mu_mn 1.04e-1 1.03e-1 6.63e-3 6.39e-3 9.34e-2 1.15e-1 1.00 1194.
3 lambda[1] 2.10e-1 1.67e-1 1.67e-1 1.36e-1 2.88e-2 5.52e-1 1.00 3855.
4 lambda[2] 1.44e-1 8.17e-2 1.70e-1 9.60e-2 4.57e-3 4.99e-1 1.00 2927.
5 lambda[3] 5.84e-2 4.76e-2 4.24e-2 3.55e-2 9.88e-3 1.41e-1 1.00 3632.
6 lambda[4] 1.43e-1 9.06e-2 1.68e-1 9.95e-2 4.92e-3 4.55e-1 0.999 3237.
7 lambda[5] 4.85e-1 4.66e-1 1.65e-1 1.56e-1 2.47e-1 7.83e-1 1.00 5490.
8 lambda[6] 2.97e-1 2.41e-1 2.32e-1 1.91e-1 4.10e-2 7.32e-1 1.00 4104.
9 lambda[7] 1.46e-1 8.97e-2 1.69e-1 9.73e-2 6.50e-3 4.79e-1 1.01 3108.
10 lambda[8] 1.45e-1 8.13e-2 1.78e-1 9.59e-2 4.25e-3 4.78e-1 1.00 2482.
# ℹ 2,993 more rows
# ℹ 1 more variable: ess_tail <num>
We have some basic HMC-based validity statistics we can check.
Show code
pnbd_long_onehiermu2_stanfit$cmdstan_diagnose()Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehiermu2-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehiermu2-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehiermu2-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_long_onehiermu2-4.csvWarning: non-fatal error reading adaptation data
Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.
Checking sampler transitions for divergences.
No divergent transitions found.
Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.
Effective sample size satisfactory.
Split R-hat values satisfactory all parameters.
Processing complete, no problems detected.
5.2 Visual Diagnostics of the Sample Validity
Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.
Show code
parameter_subset <- c(
"mu_mn",
"lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
"mu[1]", "mu[2]", "mu[3]", "mu[4]"
)
pnbd_long_onehiermu2_stanfit$draws(inc_warmup = FALSE) |>
mcmc_trace(pars = parameter_subset) +
expand_limits(y = 0) +
labs(
x = "Iteration",
y = "Value",
title = "Traceplot of Sample of Lambda and Mu Values"
) +
theme(axis.text.x = element_text(size = 10))We also check \(N_{eff}\) as a quick diagnostic of the fit.
Show code
pnbd_long_onehiermu2_stanfit |>
neff_ratio(pars = c("lambda", "mu")) |>
as.numeric() |>
mcmc_neff() +
ggtitle("Plot of Parameter Effective Sample Sizes")5.3 Assess the Model
As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.
Show code
pnbd_long_onehiermu2_assess_data_lst <- run_model_assessment(
model_stanfit = pnbd_long_onehiermu2_stanfit,
insample_tbl = customer_fit_stats_tbl,
outsample_tbl = customer_valid_stats_tbl,
fit_label = "pnbd_long_onehiermu2",
fit_end_dttm = use_fit_end_date |> as.POSIXct(),
valid_start_dttm = use_valid_start_date |> as.POSIXct(),
valid_end_dttm = use_valid_end_date |> as.POSIXct(),
sim_seed = 4210
)
pnbd_long_onehiermu2_assess_data_lst |> glimpse()List of 3
$ model_simstats_filepath : 'glue' chr "data/pnbd_long_onehiermu2_assess_model_simstats_tbl.rds"
$ model_fit_simstats_filepath : 'glue' chr "data/pnbd_long_onehiermu2_assess_fit_simstats_tbl.rds"
$ model_valid_simstats_filepath: 'glue' chr "data/pnbd_long_onehiermu2_assess_valid_simstats_tbl.rds"
5.3.1 Check In-Sample Data Validation
We first check the model against the in-sample data.
Show code
simdata_tbl <- pnbd_long_onehiermu2_assess_data_lst |>
use_series(model_fit_simstats_filepath) |>
read_rds()
insample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_fitdata_tbl,
simdata_tbl = simdata_tbl
)
insample_plots_lst$multi_plot |> print()Show code
insample_plots_lst$total_plot |> print()Show code
insample_plots_lst$quant_plot |> print()This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.
5.3.2 Check Out-of-Sample Data Validation
We now repeat for the out-of-sample data.
Show code
simdata_tbl <- pnbd_long_onehiermu1_assess_data_lst |>
use_series(model_valid_simstats_filepath) |>
read_rds()
outsample_plots_lst <- create_model_assessment_plots(
obsdata_tbl = obs_validdata_tbl,
simdata_tbl = simdata_tbl
)
outsample_plots_lst$multi_plot |> print()Show code
outsample_plots_lst$total_plot |> print()Show code
outsample_plots_lst$quant_plot |> print()As for our short time frame data, overall our model is working well.
6 Compare Model Outputs
We have looked at each of the models individually, but it is also worth looking at each of the models as a group.
Show code
calculate_simulation_statistics <- function(file_rds) {
simdata_tbl <- read_rds(file_rds)
multicount_cust_tbl <- simdata_tbl |>
filter(sim_tnx_count > 0) |>
count(draw_id, name = "multicust_count")
totaltnx_data_tbl <- simdata_tbl |>
count(draw_id, wt = sim_tnx_count, name = "simtnx_count")
simstats_tbl <- multicount_cust_tbl |>
inner_join(totaltnx_data_tbl, by = "draw_id")
return(simstats_tbl)
}Show code
obs_fit_customer_count <- obs_fitdata_tbl |>
filter(tnx_count > 0) |>
nrow()
obs_valid_customer_count <- obs_validdata_tbl |>
filter(tnx_count > 0) |>
nrow()
obs_fit_total_count <- obs_fitdata_tbl |>
pull(tnx_count) |>
sum()
obs_valid_total_count <- obs_validdata_tbl |>
pull(tnx_count) |>
sum()
obs_stats_tbl <- tribble(
~assess_type, ~name, ~obs_value,
"fit", "multicust_count", obs_fit_customer_count,
"fit", "simtnx_count", obs_fit_total_count,
"valid", "multicust_count", obs_valid_customer_count,
"valid", "simtnx_count", obs_valid_total_count
)
model_assess_tbl <- dir_ls("data", regexp = "pnbd_long_(one|fixed).*_assess") |>
enframe(name = NULL, value = "file_path") |>
filter(str_detect(file_path, "_assess_model_", negate = TRUE)) |>
mutate(
model_label = str_replace(file_path, "data/pnbd_long_(.*?)_assess_.*", "\\1"),
assess_type = if_else(str_detect(file_path, "_assess_fit_"), "fit", "valid"),
sim_data = map(
file_path, calculate_simulation_statistics,
.progress = "calculate_simulation_statistics"
)
)
model_assess_tbl |> glimpse()Rows: 14
Columns: 4
$ file_path <fs::path> "data/pnbd_long_fixed1_assess_fit_simstats_tbl.rds", …
$ model_label <chr> "fixed1", "fixed1", "fixed2", "fixed2", "fixed3", "fixed3"…
$ assess_type <chr> "fit", "valid", "fit", "valid", "fit", "valid", "fit", "va…
$ sim_data <list> [<tbl_df[2000 x 3]>], [<tbl_df[2000 x 3]>], [<tbl_df[2000…
Show code
model_assess_summstat_tbl <- model_assess_tbl |>
select(model_label, assess_type, sim_data) |>
unnest(sim_data) |>
pivot_longer(
cols = !c(model_label, assess_type, draw_id)
) |>
group_by(model_label, assess_type, name) |>
summarise(
.groups = "drop",
mean_val = mean(value),
p10 = quantile(value, 0.10),
p25 = quantile(value, 0.25),
p50 = quantile(value, 0.50),
p75 = quantile(value, 0.75),
p90 = quantile(value, 0.90)
)
model_assess_summstat_tbl |> glimpse()Rows: 28
Columns: 9
$ model_label <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed2", "fixed2"…
$ assess_type <chr> "fit", "fit", "valid", "valid", "fit", "fit", "valid", "va…
$ name <chr> "multicust_count", "simtnx_count", "multicust_count", "sim…
$ mean_val <dbl> 636.0455, 7273.8900, 53.4495, 606.6580, 698.7080, 4186.950…
$ p10 <dbl> 618.0, 6635.9, 49.0, 525.0, 680.0, 3871.0, 36.0, 276.0, 59…
$ p25 <dbl> 626.00, 6914.75, 51.00, 563.00, 689.00, 4015.00, 39.00, 30…
$ p50 <dbl> 636.0, 7248.0, 53.0, 604.0, 699.0, 4179.0, 41.0, 344.5, 61…
$ p75 <dbl> 646.00, 7592.25, 56.00, 649.00, 708.00, 4351.25, 44.00, 38…
$ p90 <dbl> 654.0, 7952.2, 58.0, 693.0, 716.0, 4511.0, 46.0, 415.0, 63…
Show code
#! echo: TRUE
ggplot(model_assess_summstat_tbl) +
geom_errorbar(
aes(x = model_label, ymin = p10, ymax = p90), width = 0
) +
geom_errorbar(
aes(x = model_label, ymin = p25, ymax = p75), width = 0, linewidth = 3
) +
geom_hline(
aes(yintercept = obs_value),
data = obs_stats_tbl, colour = "red"
) +
scale_y_continuous(labels = label_comma()) +
expand_limits(y = 0) +
facet_wrap(
vars(assess_type, name), scale = "free_y"
) +
labs(
x = "Model",
y = "Count",
title = "Comparison Plot for the Different Models"
) +
theme(
axis.text.x = element_text(angle = 20, vjust = 0.5, size = 8)
)6.1 Write Assessment Data to Disk
We now want to save the assessment data to disk.
Show code
model_assess_tbl |> write_rds("data/pnbd_long_onehier_assess_data_tbl.rds")7 R Environment
Show code
options(width = 120L)
sessioninfo::session_info()─ Session info ───────────────────────────────────────────────────────────────────────────────────────────────────────
setting value
version R version 4.2.3 (2023-03-15)
os Ubuntu 22.04.2 LTS
system x86_64, linux-gnu
ui X11
language (EN)
collate en_US.UTF-8
ctype en_US.UTF-8
tz Europe/Dublin
date 2023-06-09
pandoc 2.19.2 @ /usr/local/bin/ (via rmarkdown)
─ Packages ───────────────────────────────────────────────────────────────────────────────────────────────────────────
package * version date (UTC) lib source
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sessioninfo 1.2.2 2021-12-06 [1] RSPM (R 4.2.0)
shiny 1.7.4 2022-12-15 [1] RSPM (R 4.2.0)
shinyjs 2.1.0 2021-12-23 [1] RSPM (R 4.2.0)
shinystan 2.6.0 2022-03-03 [1] RSPM (R 4.2.0)
shinythemes 1.2.0 2021-01-25 [1] RSPM (R 4.2.0)
StanHeaders 2.21.0-7 2020-12-17 [1] RSPM (R 4.2.0)
stringi 1.7.12 2023-01-11 [1] RSPM (R 4.2.0)
stringr * 1.5.0 2022-12-02 [1] RSPM (R 4.2.0)
svUnit 1.0.6 2021-04-19 [1] RSPM (R 4.2.0)
tensorA 0.36.2 2020-11-19 [1] RSPM (R 4.2.0)
threejs 0.3.3 2020-01-21 [1] RSPM (R 4.2.0)
tibble * 3.2.1 2023-03-20 [1] RSPM (R 4.2.0)
tidybayes * 3.0.4 2023-03-14 [1] RSPM (R 4.2.0)
tidyr * 1.3.0 2023-01-24 [1] RSPM (R 4.2.0)
tidyselect 1.2.0 2022-10-10 [1] RSPM (R 4.2.0)
tidyverse * 2.0.0 2023-02-22 [1] RSPM (R 4.2.0)
timechange 0.2.0 2023-01-11 [1] RSPM (R 4.2.0)
tzdb 0.3.0 2022-03-28 [1] RSPM (R 4.2.0)
utf8 1.2.3 2023-01-31 [1] RSPM (R 4.2.0)
vctrs 0.6.2 2023-04-19 [1] RSPM (R 4.2.0)
withr 2.5.0 2022-03-03 [1] RSPM (R 4.2.0)
xfun 0.38 2023-03-24 [1] RSPM (R 4.2.0)
xtable 1.8-4 2019-04-21 [1] RSPM (R 4.2.0)
xts 0.13.1 2023-04-16 [1] RSPM (R 4.2.0)
yaml 2.3.7 2023-01-23 [1] RSPM (R 4.2.0)
zoo 1.8-12 2023-04-13 [1] RSPM (R 4.2.0)
[1] /usr/local/lib/R/site-library
[2] /usr/local/lib/R/library
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options(width = 80L)